Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight.

In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive-contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré-Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré-Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

The influence of different horseshoes and ground substrates on mid-stance hoof orientation at the walk.

BACKGROUND: Horseshoes with modified contact surfaces combined with deformable ground substrates are used to change hoof orientation during mid-stance, for example, for therapeutic reasons. OBJECTIVES: To measure the effect of horseshoes and ground substrates on sagittal and transverse plane hoof orientation at mid-stance using a dorsal hoof wall mounted triaxial accelerometer. STUDY DESIGN: In vivo experiment, randomised crossover design. METHODS: Differences in sagittal and transverse plane angles between standing and mid-stance of the left front hoof of six horses walking with regular horseshoes, egg bar, toe-wide, medial-wide, lateral-wide and three-degree egg bar shoes on turf, sand and hard ground substrates were assessed with linear mixed models with horseshoe and substrate type as fixed factors (p < 0.05) for each animal. RESULTS: Hoof angles were significantly affected by horseshoe (p < 0.001), surface (p < 0.001) and the combination (p < 0.001). The sagittal plane angle increased in deformable ground substrates at walk-in mid-stance on turf [mean (±standard deviation): 2.6° (±3.8°)] and on sand [2.6° (±4.1°)] across all shoes. The greatest increase was observed with egg bar shoes [turf: 4.37° (±3.82°); sand 4.69° (±3.83°)]. There was a tendency for the hoof to sink laterally into deformable ground substrates among all shoes [turf: 1.11° (±1.49°); sand: 0.93° (±1.93°)]. Medial-wide shoes increased the lateral sinking [turf: 2.00° (±1.63°); sand: 1.79° (±1.58°)]. Lateral-wide shoes reduced the lateral sinking on turf [0.62° (±1.26°)] and induced a marginal medial sinking on sand [-0.007° (±2.03°)]. MAIN LIMITATIONS: The substrate properties were not quantitatively assessed, and observations were limited to front hooves at the walk. A larger sample size would be preferable. CONCLUSIONS: Mid-stance hoof orientation changes with specific combinations of shoes and ground substrates in the walking horse.

CONTEXTO: É especulado que ferraduras com solados diferentes combinadas com superfícies deformáveis podem mudar a orientação do casco durante a fase de apoio, por exemplo, por razões terapêuticas. OBJETIVOS: Mensurar o efeito de diferentes ferraduras e superfícies na orientação do casco nos planos sagital e transversal durante a fase de apoio usando um acelerômetro triaxial acoplado à parte dorsal do casco. DELINEAMENTO DO ESTUDO: Experimento in vivo, delineamento randomizado e cruzado. MÉTODOS: As diferenças entre os ângulos dos planos sagital e transverso nas diferentes fases de apoio do casco do membro anterior esquerdo de seis cavalos ao passo utilizando ferradura normal, oval, oval talonada, de pinça larga e com extensão medial ou lateral na grama, areia ou superfície dura foram avaliadas utilizando modelos mistos lineares com ferradura e tipo de superfície como fatores fixos (P < 0.05) para cada animal. RESULTADOS: Os ângulos do casco foram significativamente afetados pelo tipo de ferradura (P < 0.001), superfície (P < 0.001) e pela combinação de ambos (P < 0.001). O ângulo do plano sagital aumentou em superfícies deformáveis no passo na fase de apoio na grama (média (+/­SD): 2.6 (+/−3.8) graus) e na areia (2.6 (+/−4.1 graus) para todos os tipos de ferradura. O maior aumento foi observado com a ferradura oval (grama: 4.37 (+/− 3.82) graus; areia 4.69 (+/−3.83) graus). Houve uma tendência de o casco rebaixar mais lateralmente em superfícies deformáveis com todas as ferraduras (grama: 1.11 (+/−1.49) graus; areia: 0.93 (+/−1.93) graus). Ferraduras com extensão medial aumentaram o rebaixamento lateral (grama: 2.00 (+/−1.63) graus; areia: 1.79 (+/−1.58) graus). Ferraduras com extensão lateral reduziram o rebaixamento lateral na grama (0.62 (+/−1.26) graus) e induziram o rebaixamento medial na areia (−0.007 (+/−2.03) graus). PRINCIPAIS LIMITAÇÕES: As propriedades das superfícies não foram avaliadas quantitativamente, e as observações foram limitadas aos cascos dos membros anteriores e ao passo. Um número maior de animais no estudo seria desejável. CONCLUSÕES: A orientação do casco na fase de apoio muda de acordo com combinações específicas de ferradura e superfícies no cavalo ao passo.